Lecture Notes 10: Change of Variables and Triple Integral
Ruipeng Shen
June 6, 2014
1 Change of Variable
Theorem 1 (Change of Variables). Let D and D? be elementary
regions in R2 and let T : D? D be a C1, one-to-one map such that
T(D?) = D. For any integrable function f : D R,
D
f(x, y)dA =
D?
f(x(u, v), y(u, v))
(x, y)(u, v) dA?,
where(x, y)
(u, v)= det(DT) =
x/u x/vy/u y/v = xu yv xv yu.
is called the Jacobian of T.
Example 2 (Polar Coordinates). Let T(r, ) = (r cos , r sin ) be
the mapping that defines thepolar coordinates. This is a one-to-one
map from [r1, r2] [1, 2] to its image if 0 < r1 < r2and 0 1
< 2 < 2pi. Its Jacobian is r.Definition 3 (Affine Maps). A
map T : R2 R2 defined by
T
[uv
]=
[a11 a12a21 a22
] [uv
]+
[b1b2
]is called an affine map. If b1 = b2 = 0, it is also called a
linear map.
Example 4. Calculate the Jacobian of an affine map.
Theorem 5. Let Tu = Au + b be an affine map and det(A) 6= 0.
Then(a) T is one-to-one.
(b) T maps lines to lines, parallel lines to parallel lines.
(c) T maps a parallelogram to a parallelogram.
(d) Its inverse T1u = A1uA1b is another affine map.Example 6.
Let F be an affine map and D? = [0, 1] [0, 1], f = C. Check theorem
1.Example 7. Find the integral
D
(x+y)4(xy)4dA, here D is the square {(x, y) : |x|+|y| 1}.
Example 8. Let B(0, R) = {(x, y)|x2 + y2 R2}, calculate
B(0,R)
ex2y2dA.
Example 9. Find the area of the region bounded by the x-axis,
y-axis and the spiral c(t) =(et cos t, et sin t) with t [0,
pi/2].
1
2 Triple Integral
Definition of triple integral over a rectangular box Similar to
the double integral, theintegral is defined as the limit of Riemann
sums.
Theorem 10 (Fubinis theorem). Let f(x, y, z) be an integrable
function defined on a rectangularbox W = [a1, b1] [a2, b2] [a3,
b3]. Then
W
f(x, y, z)dV =
b3a3
( b2a2
( b1a1
f(x, y, z)dx
)dy
)dz.
Furthermore, let D = [a1, b1] [a2, b2]W
f(x, y, z)dV =
D
( b3a3
f(x, y, z)dz
)dA.
Example 11. Let W = [0, 1] [0, 2] [0, 3], calculate Wz2(2x+
y)dV
Triple integral over a general region If f(x, y, z) is defined
in a general region W , we canchoose a large rectangular box W1 so
that W W1 and a new function
F (x, y, z) =
{f(x, y, z), (x, y, z) W ;0, (x, y, z) W1 \W.
and define W
f(x, y, z)dV =
W1
F (x, y, z)dV.
Type 1 region Let the region W be given by
W = {(x, y, z) R3 : (x, y) D R2, g1(x, y) z g2(x, y)}.Here D is
an elementary region in R2; The functions g1(x, y) and g2(x, y) are
continuously definedon D such that g1(x, y) g2(x, y) for each (x,
y) D. Then we have
W
f(x, y, z)dV =
D
( g2(x,y)g1(x,y)
f(x, y, z)dz
)dA.
Example 12. Calculate the triple integral
W2xdV . Here W is the tetrahedron in the first
octant bounded by x+ y + z = 3.
Theorem 13 (Change of Variables). Let W and W ? be elementary
regions in R3 and let
T = T(u, v, w) = (x(u, v, w), y(u, v, w), z(u, v, w)) : W ? Wbe
a C1, one-to-one map such that T(W ?) = W . For any integrable
function f : W R,
W
f(x, y, z)dV =
W?
f(x(u, v, w), y(u, v, w), z(u, v, w))
(x, y, z)(u, v, w) dV ?,
where
(x, y, z)
(u, v, w)= det(DT) =
x/u x/v x/wy/u y/v y/wz/u z/v z/w
is called the Jacobian of T.
Example 14. Let W be the ball {(x, y, z)|x2 + y2 + z2 1}, find
the integralW
x2 + y2 + z2dV.
Use the spherical coordinates (x, y, z) = ( sin cos , sin sin ,
cos)
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